The Outdoor Cycle workout on the watch is almost useless without the phone. (I tried it, and it pretty much captures nothing of value.) So, you need the phone if you do this workout. Running and walking work OK without the phone, but they get way better distance accuracy with the phone. I think that of the other workouts (Other, Indoor Cycle, Indoor Run, Indoor Walk, Elliptical, Rower, and Stair Stepper) are perfectly fine without the phone.
For outdoor cycling, I will always carry my phone. Safety and emergency are the reason-- getting stranded because of a mechanical failure really sucks. Plus, I just got a Garmin Edge for my bike that links directly to my phone and the Internet (through my phone) for some really cool stuff.
For running, I only carry my phone for safety on long runs and for music when I am running alone. I could do music from my AW, but it is just easier with the phone. Then I use the AW's music app to control what I am listening to.
